Global Well-Posedness of Displacement Monotone Degenerate Mean Field Games Master Equations

In this manuscript we construct global in time classical solutions to mean field games master equations in the lack of idiosyncratic noise in the individual agents' dynamics. These include both deterministic models and dynamics driven solely by a Brownian common noise. We consider a general class of non-separable Hamiltonians and final data functions that are supposed to be displacement monotone. Our main results unify and generalize in particular some of the well-posedness results on displacement monotone master equations obtained recently by Gangbo--M\'esz\'aros and Gangbo--M\'esz\'aros--Mou--Zhang.

[1]  Chenchen Mou,et al.  Minimal solutions of master equations for extended mean field games , 2023, 2303.00230.

[2]  P. J. Graber,et al.  On monotonicity conditions for Mean Field Games , 2022, 2210.02281.

[3]  P. J. Graber,et al.  On Mean Field Games and Master Equations through the lens of conservation laws , 2022, 2208.10360.

[4]  P. Souganidis,et al.  Mean field games with common noise and degenerate idiosyncratic noise , 2022, 2207.10209.

[5]  P. Souganidis,et al.  Monotone Solutions of the Master Equation for Mean Field Games with Idiosyncratic Noise , 2022, SIAM J. Math. Anal..

[6]  F. Delarue,et al.  Weak solutions to the master equation of potential mean field games , 2022, 2204.04315.

[7]  Chenchen Mou,et al.  Mean Field Game Master Equations with Anti-monotonicity Conditions , 2022, 2201.10762.

[8]  Alp'ar R. M'esz'aros,et al.  Mean Field Games systems under displacement monotonicity , 2021, 2109.06687.

[9]  A. M'esz'aros,et al.  Well-posedness of mean field games master equations involving non-separable local Hamiltonians , 2021, 2105.03926.

[10]  P. Lions,et al.  Dimension reduction techniques in deterministic mean field games , 2021, Communications in Partial Differential Equations.

[11]  W. Gangbo,et al.  Mean field games master equations with nonseparable Hamiltonians and displacement monotonicity , 2021, The Annals of Probability.

[12]  P. Souganidis,et al.  On first order mean field game systems with a common noise , 2020, The Annals of Applied Probability.

[13]  C. Bertucci,et al.  Monotone solutions for mean field games master equations: finite state space and optimal stopping , 2020, Journal de l’École polytechnique — Mathématiques.

[14]  A. Bensoussan,et al.  Control on Hilbert Spaces and Application to Some Mean Field Type Control Problems , 2020, 2005.10770.

[15]  W. Gangbo,et al.  Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games , 2020, Communications on Pure and Applied Mathematics.

[16]  Pierre Cardaliaguet,et al.  Splitting methods and short time existence for the master equations in mean field games , 2020, Journal of the European Mathematical Society.

[17]  Wilfrid Gangbo,et al.  On differentiability in the Wasserstein space and well-posedness for Hamilton–Jacobi equations , 2019, Journal de Mathématiques Pures et Appliquées.

[18]  Chenchen Mou,et al.  Wellposedness of Second Order Master Equations for Mean Field Games with Nonsmooth Data , 2019, Memoirs of the American Mathematical Society.

[19]  Sergio Mayorga.  Short time solution to the master equation of a first order mean field game , 2018, Journal of Differential Equations.

[20]  Kavita Ramanan,et al.  From the master equation to mean field game limit theory: Large deviations and concentration of measure , 2018, The Annals of Probability.

[21]  K. Ramanan,et al.  From the master equation to mean field game limit theory: a central limit theorem , 2018, Electronic Journal of Probability.

[22]  R. Carmona,et al.  Probabilistic Theory of Mean Field Games with Applications II: Mean Field Games with Common Noise and Master Equations , 2018 .

[23]  François Delarue,et al.  Probabilistic Theory of Mean Field Games with Applications I: Mean Field FBSDEs, Control, and Games , 2018 .

[24]  Jianfeng Zhang,et al.  Backward Stochastic Differential Equations , 2017 .

[25]  Jianfeng Zhang,et al.  An Elementary Proof for the Structure of Wasserstein Derivatives , 2017 .

[26]  Tzu-Wei Yang,et al.  Forward–backward stochastic differential equations with monotone functionals and mean field games with common noise , 2016, Stochastic Processes and their Applications.

[27]  Wilfrid Gangbo,et al.  Existence of a solution to an equation arising from the theory of Mean Field Games , 2015 .

[28]  P. Lions,et al.  The Master Equation and the Convergence Problem in Mean Field Games , 2015, 1509.02505.

[29]  D. Crisan,et al.  A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria , 2014, Memoirs of the American Mathematical Society.

[30]  R. Carmona,et al.  Mean field games with common noise , 2014, 1407.6181.

[31]  Saran Ahuja,et al.  Wellposedness of Mean Field Games with Common Noise under a Weak Monotonicity Condition , 2014, SIAM J. Control. Optim..

[32]  R. Carmona,et al.  Forward-Backward Stochastic Differential Equations and Controlled McKean Vlasov Dynamics , 2013, 1303.5835.

[33]  Marc Quincampoix,et al.  Lipschitz continuity and semiconcavity properties of the value function of a stochastic control problem , 2010 .

[34]  P. Cannarsa,et al.  Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control , 2004 .

[35]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .

[36]  P. Cardaliaguet,et al.  An Introduction to Mean Field Game Theory , 2020 .

[37]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[38]  Jules Michelet,et al.  Cours au Collège de France , 1995 .